#7627568

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3542	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,23)(17,22)(18,24)(19,25)(21,26)(27,28);(5,6,8,10)(7,9,12,11)(13,15)(14,20)(16,22)(17,23)(18,26)(19,27)(21,24)(25,28);(5,6)(7,9)(8,10)(11,12)(13,16)(14,21)(15,18)(17,20)(19,26)(22,27)(23,25)(24,28);(5,7)(6,9)(8,12)(10,11)(13,17,19,24)(14,22,25,18)(15,21,27,23)(16,20,26,28);(1,3,4,2)(13,16)(14,23)(15,22)(17,20)(18,27)(19,26)(21,25)(24,28);(13,18,19,22)(14,24,25,17)(15,26,27,16)(20,21,28,23)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236840, #72238148, #72238254
maximal normal subgroup of	#72236840, #72238148, #72238254
maximal quotient of	#72263775, #72263797

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534764);